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*Please note the room change for this talk.*

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We consider a Bayesian framework for making inferences about dynamicalÌýsystems from ergodic observations.The proposed Bayesian procedure isÌýbased on the Gibbs posterior, a decision theoretic generalization ofÌýstandard Bayesian inference. We place a prior over a model classÌýconsisting of a parametrized family of Gibbs measures on a mixingÌýshift of finite type. This model class generalizes (hidden) MarkovÌýchain models by allowing for long range dependencies, including MarkovÌýchains of arbitrarily large orders. We characterize the asymptoticÌýbehavior of the Gibbs posterior distribution on the parameter space asÌýthe number of observations tends to infinity. In particular, we defineÌýa limiting variational problem over the space of joinings of the modelÌýsystem with the observed system, and we show that the Gibbs posteriorÌýdistributions concentrate around the solution set of this variationalÌýproblem.In the case of properly specified models our convergenceÌýresults may be used to establish posterior consistency. This workÌýestablishes tight connections between Gibbs posterior inference andÌýthe thermodynamic formalism, which may inspire new proof techniques inÌýthe study of Bayesian posterior consistency for dependent processes.
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Time permitting a discussion of large deviation principles,specifically the Laplace principle, as general tool for provingÌýposterior consistency.